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Fixed points theorems for $b$-enriched multivalued nonexpansive mappings and *-$b$-enriched nonexpansive mappings

Ioan Trifoi

TL;DR

The paper addresses extending fixed-point theory from single-valued $b$-enriched nonexpansive mappings to the multivalued setting in Hilbert spaces by introducing *-$b$-enriched nonexpansive mappings. It demonstrates that averaging with $T_b=(1-b)I+b T$ and $b= frac{1}{b+1}$ preserves fixed points, i.e., $F(T)=F(T_b)$, and develops Krasnoselskii-type iterations to approximate these fixed points. The authors prove Browder-type existence and convexity results, and establish both strong and weak convergence of these iterative schemes under conditions like hemicompactness and weak compactness, respectively. Collectively, the work generalizes fixed-point results to enriched multis valued operators and provides a rigorous iterative framework for finding fixed points in Hilbert spaces, laying groundwork for extensions to other spaces and more general multivalued operators.

Abstract

The main purpose of this paper is to extend some fixed point results for single valued $b$-enriched nonexpansive mappings to the case of multivalued mappings. To this end, we introduce *-$b$-enriched nonexpansive mappings, as a generalization of *-nonexpansive mappings \cite{Abdul Rahim Khan} for which we establish an existence theorem in Hilbert space. We proved weak and strong convergence results of Krasnoselskii iteration process for $b$-enriched multivalued nonexpasive mappings and *-$b$-enriched nonexpansive mappings.

Fixed points theorems for $b$-enriched multivalued nonexpansive mappings and *-$b$-enriched nonexpansive mappings

TL;DR

The paper addresses extending fixed-point theory from single-valued -enriched nonexpansive mappings to the multivalued setting in Hilbert spaces by introducing *--enriched nonexpansive mappings. It demonstrates that averaging with and preserves fixed points, i.e., , and develops Krasnoselskii-type iterations to approximate these fixed points. The authors prove Browder-type existence and convexity results, and establish both strong and weak convergence of these iterative schemes under conditions like hemicompactness and weak compactness, respectively. Collectively, the work generalizes fixed-point results to enriched multis valued operators and provides a rigorous iterative framework for finding fixed points in Hilbert spaces, laying groundwork for extensions to other spaces and more general multivalued operators.

Abstract

The main purpose of this paper is to extend some fixed point results for single valued -enriched nonexpansive mappings to the case of multivalued mappings. To this end, we introduce *--enriched nonexpansive mappings, as a generalization of *-nonexpansive mappings \cite{Abdul Rahim Khan} for which we establish an existence theorem in Hilbert space. We proved weak and strong convergence results of Krasnoselskii iteration process for -enriched multivalued nonexpasive mappings and *--enriched nonexpansive mappings.

Paper Structure

This paper contains 2 sections, 15 theorems, 95 equations.

Table of Contents

  1. Main results
  2. Conclusions

Key Result

Theorem 1

Mujahid Abbas Rizwan Anjum Vasile Berinde Let $X$ a uniformly convex Banach space and $D$ a closed convex bounded nonempty subset of $X$. Let $T:D \rightarrow C(D)$ be a $b$-enriched multivalued nonexpansive mapping. Then $T$ has a fixed point (i.e. there exist $x\in D$ with $x \in Tx$ )

Theorems & Definitions (37)

  • Definition 1: Vasile Berinde 2010
  • Definition 2
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • ...and 27 more